{"id":9055,"date":"2023-04-27T10:00:03","date_gmt":"2023-04-27T04:30:03","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=9055"},"modified":"2023-04-28T17:41:15","modified_gmt":"2023-04-28T12:11:15","slug":"proofs-similar-triangles","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/proofs-similar-triangles\/","title":{"rendered":"Proofs with Similar Triangles"},"content":{"rendered":"<h2><span style=\"color: #00ccff;\">Proofs with <a href=\"https:\/\/www.aplustopper.com\/similarity-of-triangles\/\">Similar Triangles<\/a><\/span><\/h2>\n<p><strong>Definition:<\/strong><br \/>\nTwo triangles are <strong>similar<\/strong> if and only if the corresponding sides are in proportion and the corresponding angles are congruent.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-124036\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-1.png\" alt=\"Proofs with Similar Triangles 1\" width=\"760\" height=\"438\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-1.png 760w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-1-300x173.png 300w\" sizes=\"(max-width: 760px) 100vw, 760px\" \/><br \/>\n<strong>There are three accepted methods of proving triangles similar:<\/strong><\/p>\n<h3><span style=\"color: #0000ff;\">AA<\/span><\/h3>\n<p>To show two triangles are similar, it is sufficient to show that two angles of one triangle are congruent (equal) to two angles of the other triangle.<br \/>\n<strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-124037 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-2.jpg\" alt=\"Proofs with Similar Triangles 2\" width=\"558\" height=\"126\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-2.jpg 558w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-2-300x68.jpg 300w\" sizes=\"(max-width: 558px) 100vw, 558px\" \/>Theorem:<\/strong><br \/>\nIf two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.<\/p>\n<h3><span style=\"color: #0000ff;\">SSS for similarity<\/span><\/h3>\n<p>SSS for similar triangles is NOT the same theorem as we used for congruent triangles. To show triangles are similar, it is sufficient to show that the three sets of corresponding sides are in proportion.<br \/>\n<strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-124039 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-3.jpg\" alt=\"Proofs with Similar Triangles 3\" width=\"567\" height=\"165\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-3.jpg 567w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-3-300x87.jpg 300w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/>Theorem:<\/strong><br \/>\nIf the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.<\/p>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/JWattj1ERrA?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3><span style=\"color: #0000ff;\">SAS for similarity<\/span><\/h3>\n<p>SAS for similar triangles is NOT the same theorem as we used for congruent triangles. To show triangles are similar, it is sufficient to show that two sets of corresponding sides are in proportion and the angles they include are congruent.<br \/>\n<strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-124040 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-4.jpg\" alt=\"Proofs with Similar Triangles 4\" width=\"562\" height=\"154\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-4.jpg 562w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-4-300x82.jpg 300w\" sizes=\"(max-width: 562px) 100vw, 562px\" \/>Theorem:<\/strong><br \/>\nIf an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.<\/p>\n<p><strong>Once the triangles are similar:<\/strong><br \/>\n<strong>Theorem:<\/strong><br \/>\nThe corresponding sides of similar triangles are in proportion.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-124041 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-5.jpg\" alt=\"Proofs with Similar Triangles 5\" width=\"574\" height=\"125\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-5.jpg 574w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-5-300x65.jpg 300w\" sizes=\"(max-width: 574px) 100vw, 574px\" \/><\/p>\n<p><strong>Dealing with overlapping triangles:<\/strong><br \/>\nMany problems involving similar triangles have one triangle ON TOP OF (overlapping) another triangle.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-124042 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-6.jpg\" alt=\"Proofs with Similar Triangles 6\" width=\"246\" height=\"193\" \/>Since \\(\\overline { DE\u00a0}\\)\u00a0is marked to be parallel to \\(\\overline { AC }\\), we know that we have \u2220BDE congruent to \u2220DAC (by corresponding angles). \u2220B is shared by both triangles, so the two triangles are similar by AA.<\/p>\n<p>There is an additional theorem that can be used when working with overlapping triangles:<br \/>\n<strong>Additional Theorem:<\/strong> If a line is parallel to one side of a triangle and intersects the other two sides of the triangle, the line divides these two sides proportionally.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-124043 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Proofs-with-Similar-Triangles-7.jpg\" alt=\"Proofs with Similar Triangles 7\" width=\"181\" height=\"101\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Proofs with Similar Triangles Definition: Two triangles are similar if and only if the corresponding sides are in proportion and the corresponding angles are congruent. There are three accepted methods of proving triangles similar: AA To show two triangles are similar, it is sufficient to show that two angles of one triangle are congruent (equal) [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[3314,3313,3316,3315],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Proofs with Similar Triangles - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/proofs-similar-triangles\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Proofs with Similar Triangles\" \/>\n<meta property=\"og:description\" content=\"Proofs with Similar Triangles Definition: Two triangles are similar if and only if the corresponding sides are in proportion and the corresponding angles are congruent. 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